On Density and Hypercyclicity

Transcription

1 International Mathematical Forum, Vol. 8, 2013, no. 9, HIKARI Ltd, On Density and Hypercyclicity Parvin Karami Department of Mathematics Islamic Azad University, Branch of Mamasani P.O.Box , Mamasani, Iran Mezban Habibi Department of Mathematics Dehdasht Branch, Islamic Azad University, Dehdasht, Iran P. O. Box , Lidingo, Stockholm, Sweden Fatemeh Safari department of mathematics Islamic Azad University, branch of Mamasani P.O.Box , Mamasani, Iran Mohammad Zarrabi department of mathematics Islamic Azad University, branch of Mamasani P.O.Box , Mamasani, Iran Copyright c 2013 Parvin Karami, Mezban Habibi, Fatemeh Safari and Mohammad Zarrabi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The aim of the paper ahead Birhoff, Maclane, Godefroy-Shapiro and

2 402 P. Karami, M. Habibi, F. Safari and M. Zarrabi Kitai-Getner-Shapiro and the results of their theorems and hypercyclic operators on space H(C). In Birhoffs theorem is shown that, if b is non-zero, then the shift with the vector b is an hypercyclic operator. Maclane in 1952 showed that the Differentiation operator on H(C) is an hypercyclic operator. Bourdon and Shapiro also studied the behavior of composition operators on this space. Mathematics Subject Classification: 47A16, 47B37 Keywords: Hypercyclicity criterion, Hypercyclic vector, Density, Independently 1 Introduction Let X be a Frechet space and T be a bounded linear operator on X. For each x X put Orb(T,x)={T n (x) :n 0)} = {x, T x, T 2 x, T 3 x,...} The set Orb(T,x) is called orbit of vector x under the operator T and the operator T is called hypercyclic operator if there exist vector x in X such that the set Orb(T,x) is dense in X, in this case the vector x is called hypercyclic vector for the operator. If X be the dual space of space X and both operators T : X X and T : X X are hypercyclic, then the operator T is called dual hypercyclic. For more information readers can see [1 5]. 2 Preliminary Notes Suppose that H(C) be the space of all functions of one complex variable with the uniform convergence topology on compact subsets of C. Consider Banach space E, Ferechet algebra by elements of dual space with uniform convergence topology over the balls of E. Space H bc (E) containing all bounded functions on compact subsets of E, the space H bc (E) includes all functions f = n=0 P n in which P n Span{ϕ n : ϕe }, n =0, 1, 2, 3,... P n 1 n =(Sup x <1 P n ) 1 n 0, n The operator φ : H(E) H(E) by definition φ(f) =Df is called differentiation operator. Let ϕ H(C), then the operator C ϕ : H(C) H(C) by definition C ϕ (f) =foϕ on H(C) is hypercyclic, if and only if the operator ϕ

3 On density and hypercyclicity 403 is a shift with a non-zero vector b C. In other words, 0 b C, ϕ(z) = z+b(see[1]). Differentiation operator on H(C) is a hypercyclic operator(see[6]). If φ(z) = α 0 C α z α be non-constant entire function on C, Then the operator φ D : H(C n ) H(C n ) by definition φ D (f) = α 0 C α D α f,f H(C) is hypercyclic operator(see[10]). Also all continuous linear operator on H(C n ) substitute with translation, if and only if, be for one ϕ H(C n ) is of exponential form T = ϕ D. Theorem 2.1 (Hypercyclicity Criterion) Let X be an F -space and T : X X be a continuous linear operator and assume that U, V are two dense subsets of X and {n k } k=1 be a sequence of positive integers, and there are sequences S nk : V X of mapping such that, (1). T n k 0,k, P ointwise on U (2). S nk 0,k, P ointwise on V (3). T n k Snk = I V then the operator T is hypercyclic 3 Main Results Theorem 3.1 If E be a Banach Space then the collection B = {e ϕ : ϕ E } is an independently linear subset of H bc (E). Theorem 3.2 Let U be an open subset of E, then S = Span{e ϕ : ϕ U} is a dense subset of H bc (E). Proof. Let ϕ 0 E and Λ : H bc (E) H bc (E) byλ(ψ) =e ϕ 0.ψ. Suppose ψ 1,ψ 2 H bc (E) and Λ(ψ) 1 =Λ(ψ) 2,soe ϕ 0.ψ 1 = e ϕ 0.ψ 2. Since e ϕ0 0, then ψ 1 = ψ 2, that is the operator Λ is one-one operator. Since constant operator and identity operator are continuous, then the operator Λ is continuous. Now since Λ(ψ) 1 = e ϕ 0.ψ is continuous operator, then the operator Λ is a homeomorphism and Span{e ϕ 0+ϕ : ϕ U} = H bc (E) Span{e +ϕ : ϕ U} = H bc (E) If λ o U then take U 0 = {ϕ λ 0 : ϕ U}, then 0 = λ 0 λ 0 U 0, So without lost of generality we can suppose 0 U. If U be a non-empty open subset of E, such that the norm of all element in U are not zero, then theorem is trivial. So assume that ϕ 0 U, ϕ 0 0 and define U 0 = { 1 ϕ : ϕ U}. ϕ 0 Now we have 1 ϕ 0 ϕ 0 = 1 ϕ 0. ϕ 0 =1, 1 ϕ 0 ϕ 0 U 0.

4 404 P. Karami, M. Habibi, F. Safari and M. Zarrabi So we have an open non-empty subset of E contain an element of norm 1. Now take δ>0 such that, U = {ϕ E : ϕ <δ}. Specially, for 0 U we have 1 U, Now we just to proof that, ϕ n S, n 0, ϕ U For this, suppose that ϕ n U for ϕ n U and n k 1. In this way we have ψ t = etϕ 1 tϕ (tϕ)2... (tϕ)k 2! t k Since tϕ U, assume that x E be given, then (ψ t ϕk )(x) = 1 t k (etϕ 1 tϕ (tϕ)2 2! t n k+1 Then in the space H bc (E) we have n k 1 ϕ(x) n t n! te δ x... (tϕ)k )(x) ψ t ϕk, t So ϕk S, and by this the proof is complete. Theorem 3.3 Let T : X X be a hypercyclic operator and U : X Y be a one by one operator with the dense range, then UTU 1 : Y Y is hypercyclic. Proof. Take hypercyclic vector x X, so we have to show that U(x) Y is a hypercyclic vector for UTU 1. Since U(x) Y is a hypercyclic vector for UTU 1 then Orb(UTU 1,U(x)) = {(UTU 1 ) n (U(x)) : n =1, 2, 3,...} = {UT n U 1 (U(x)) : n =1, 2, 3,...} = {UT n (U 1 (U(x))) : n =1, 2, 3,...} = {UT n (x) :n =1, 2, 3,...} = U({T n (x) :n =1, 2, 3,...}) = U(Orb(T,x)) so Orb(UTU 1,U(x)) = U(Orb(T,x)) = Y.

5 On density and hypercyclicity 405 Since T (A) T (A) then U(X) =U(Orb(T,x)) U(Orb(T,x) Y so Y = U(X) U(Orb(T,x) Y, now we have U(Orb(T,x)=Y, in other hand (UTU 1,U(x)) = Y. This concluded that the vector U(x) Y is a hypercyclic operator for UTU 1. ACKNOWLEDGEMENTS. This research was partially supported by a grant from Research Council of Islamic Azad University(IAU), Mamasani Branch, so the authors gratefully acknowledge this support. References [1] B. Bourdon, Invariant Manifolds of Hypercyclic Vectors, Proc. Amer. Math. Soc., 118, No. 3 (1993), [2] D. A. Herrero, Hypercyclic Operators and Chaos,Jour. of Op. Theory, 28 (1992), [3] C. Kitai, Invariant Closed Sets for Linear Operators,Thesis, University of Toronto (1982). [4] G. R. MacLane, Sequences of derivatives and normal families, Jour. Anal. Math., 2 (1952), [5] J. H. Shapiro, Composition Operators and classical function theory, Springer-Verlag, Received: November, 2012